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Origin of linear and nonlinear giant magnetoimpedance
PHYgCAL ELSEVIER
Physica A 241 (1997) 439-443
Origin of linear and nonlinear giant magnetoimpedance A. Yelon*, M. Britel, D. Menard, P. Ciureanu Groupe de Recherche en Physique et Technologie de Couches Minces and Department of Engineering Physics, Ecole Polytechnique, P.O. Box 6079, Station "Centre Ville", Montreal H3C 3A7, Canada
Abstract
It has gradually become clear that giant magnetoimpedance, which initially appeared very mysterious, can be explained classically, as arising from the dependence of skin depth upon the magnetic permeability. As the permeability may vary with dc field or with frequency, the impedance will vary as a function of these parameters. Thus, a variety of circumstances may give rise to GMI. A number of these circumstances are considered. The equivalence of GMI and ferromagnetic resonance calculations is discussed. On the basis of this equivalence, some effects are predicted.
PACS: 72.15.-v; 75.15.Nj; 76.50.+g Keywords: Giant magnetoimpedance; Ferromagnetic resonance; Metal permeability
During its brief history [ 1-6] the phenomenon of giant magnetoimpedance (GMI) has aroused considerable interest, associated with the phenomenon itself, and with its potential applications [6]. The reasons for this interest are easily understood. If we consider a magnetic fiber, wire, or ribbon, to which we apply an alternating current, its impedance is a function of the applied field and of the frequency and amplitude of the current,
Z(H, co,I ) = R + j X .
(1)
The magnetoimpedance may be defined as ~(H,~,I)=
I Z I ( H ) - ]Zl(Hr)
IZl(Hr)
(2) '
is the reference field, usually zero field. In the MHz frequency range, for very soft magnetic materials, positive magnetoimpedance in excess of 100%, [4,7] and w h e r e Hr
* Corresponding author. Fax: 514 340 3218; e-mail: [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 1 2 1-0
A. Yelon et al./Physica A 241 (1997) 439-443
440
negative magnetoimpedance of as much as 70% [2,8] have been observed at low and high values of H, respectively. It has been recognized [2,9,10] that these large changes are due to the change in skin depth arising from the change in magnetic permeability of the material with applied magnetic field. Since the ac current flows in the skin depth, the impedance increases as the permeability increases [11],
~=[
pc2 ] 1/2 L2x~o/~tj
(3)
,
where p is the resistivity, and ~t is the transverse permeability. In order for the effect to be appreciable, the sample must be much thicker than 6 (however, it has been observed in multilayer film structures [6]). In that case [11], Z=(1
- J)2-F~"pL
_
(1 _._~J)L(2~PC°I~t)'/2
(4)
In Eq. (4), L is the length of the sample, and [ is its width, for a ribbon, or a dimension of the order of the circumference for a fiber or wire. We see from Eq. (4) that GMI is favored by large values of p and ~o, and by large changes of #t with H. Thus, the GMI problem is the problem of identifying and quantifying the circumstances in which these conditions, especially the last, are met. In order to investigate these conditions, it is useful to draw a distinction between linear and nonlinear impedance and linear and nonlinear GMI effects. By linear impedance, we mean that Z is independent of I, up to some maximum value, as shown in Fig. 1. Thus, the response to sinusoidal current is a sinusoidal voltage, as shown in Fig. 2(a). This behavior is typical of observations at large values of H, in which the static magnetization is saturated. At fields near, or below those necessary to reverse the magnetization, the typical behavior will be nonlinear, as shown in Fig. 2(b). Typically, linear phenomena have been investigated before nonlinear phenomena, which may be expected to be more complicated. In the development of GMI studies, 10 ........
8-
f
I-+-,0o k.il
........
[
........
A
C~ 6 N
4-
20 0.01
I!
i I
I I
0.1
1
10
I rms (mA)
Fig. 1. Amplitude of the impedance, Z, as a function of the amplitude of the ac current, measured on a 125 lam, as quenched, CoFeSiB wire, at H = 0 , f = 100 kHz.
A. Yelon et al. IPhysica A 241 (1997) 439-443
441
b)
a)
Fig. 2. Current and voltage vs time, measuredon the same sample as for Fig. 1, at lrms= 10mA, f = 1MHz. (a) H = 40 Oe. The current and voltage waveforms are both sinusoidal. (b) H = 0. The current waveform is sinusoidal, but the voltage waveformis not.
it has been the low H, nonlinear effects which have received more attention than the high H, linear effect, despite the fact that they are indeed more complicated. This has been driven by the interest in the extremely high sensitivity obtainable for sensors, or other applications, using low field, positive, GMI. The models which have been used to obtain /tt have frequently been quasistatic in nature. For example, critical curves were used by Beach and colleagues [12] for studying plated wire, and by Humphrey [13] for solid magnetic wire. Machado and Rezende [10] used a domain switching model to treat the behavior of ribbons. These calculations give a good account of experiment up to a few MHz, but it is clear that it is preferable to use a fully dynamic model. Beach and Berkowitz [9] have presented a phenomenological dynamic approach to GMI, and Panina and colleagues [7] have recently developed an expression for #t which takes account of magnetic damping. However, we have recently pointed out [14] that a rigorous theoretical framework which takes account of both magnetic damping and of the exchange conductivity effect was developed over 40 years ago [15,16]. The general theoretical problem of GMI is the determination of the impedance from Maxwell's equations and the Landau-Lifschitz equation, taking account of the conductivity and of the detailed magnetic structure. In the problem of ferromagnetic resonance (FMR) in metals, one needs to find the surface impedance [15] rather than the bulk impedance. However, it is very simple to show [14] that as long as the sample length
442
A. Yelon et al./Physica A 241 (1997) 439-443
is shorter than the electromagnetic wavelength, the two impedances are directly proportional to each other. Since the magnetoimpedance involve a ratio, predictions may be obtained directly from the FMR literature. Since the great majority of FMR experiments are performed at high, saturating, values of H and at low ac fields, there is an abundant literature on high field, linear, calculations and their predictions for metallic samples [17]. In addition, the behavior of metallic samples has been shown not to differ greatly from that of nonmetallic samples (that is, the exchange-conductivity effect modifies the behavior only slightly). Thus for cases for which a rigorous solution is not available, results for insulators yield a good first approximation of what should be expected. There also exist [18] calculations of FMR in multidomain samples which should be useful for those who wish to treat the nonlinear GMI effects observed under these circumstances. Recognition of the simple relationship between GMI and FMR permits us to make predictions for GMI behavior in simple circumstances for which the results are well known. For example, at high fields, the imaginary part of Z should be small, except near FMR. Large values of X at low frequency indicate non-saturation within the skin depth. The dependence of Z on frequency should follow the variation of the real part, as we have observed for permalloy wires [14]. The case of longitudinal GMI (H parallel to 1) in ribbons corresponds exactly to the problem of parallel resonance in plates and films. In fibers and wires, the behavior should be similar. Transverse GMI in a ribbon will be very different, depending upon the orientation of H. H perpendicular to the ribbon corresponds essentially to the problem of perpendicular resonance in plates and films, whereas there should not be a response at large H and small 1, when H is in the plane of the ribbon. However, at large I, there must be a nonlinear response, corresponding to the problem of parallel pumping in FMR [19]. Transverse GMI in fibers and wires must show a relatively small, broad peak, since different sections of the wire have different orientation of H with respect to the radius. What is the future of GMI research, now that the underlying pheonmena are reasonably well understood? Clearly, the range of possible applications is vast [6], and much work needs to be done to optimize materials and geometries for these applications. We also foresee the potential for GMI to become a technique for the study of magnetic materials whose importance should be comparable to, or even exceed, that of FMR. References [1] L.V. Panina, K. Mohri, Appl. Phys. Lett. 65 (1994) 1189. [2] L.V. Panina, K. Mohri, K, Bushida, M. Noda, J. Appl. Phys. 76 (1994) 6198. [3] R.S. Beach, A.E. Berkowitz, Appl. Phys. Lett. 64 (1994) 3652. [41 K.V. Rao, F.B. Humphrey, J.L. Costa-Krfimer, J. Appl. Phys. 76 (1994) 6204. [5] M. Knobel, M.L. Sanchez, J. Velazquez, M. Vasquez, J. Phys.: Condens. Matter 7 (1995) Ll15. [61 L.V. Panina, K. Mohri, T. Uchiyama, Physica A 241 (1997) 429, these Proceedings. [7] L.V. Panina, K. Mohri, T. Uchiyama, M. Noda, K. Bushida, IEEE Trans. Magn. 31 (1995) 1249. [8] P. Ciureanu, P. Rudkowski, G. Rudkowska, D. Mrnard, M. Britel, J.F. Currie, J.O. StriSm-O|sen, A. Yelon, J. Appl. Phys. 79 (1996) 5136.
A. Yelon et al. IPhysica A 241 (1997) 43~443
[9] R.S. Beach, A.E. Berkowitz, J. Appl. Phys. 76 (1994) 6209. [10] F.L.A. Machado, S.M. Rezende, J. Appl. Phys. 79 (1996) 6558. [11] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [12] R.S. Beach, N. Smith, C.L. Platt, F. Jeffers, A.E. Berkowitz, Appl. Phys. Lett. 68 (1996) 2753. [13] F.B. Humphrey, private communication. [14] A. Yelon, D. Menard, M. Britel, P. Ciureanu, Appl. Phys. Lett. 69 (1996) 3084. [15] W.S. Ament, G.T. Rado, Phys. Rev. 97 (1955) 1558. [16] J.R. MacDonald, Phys. Rev. 103 (1956) 280. [17] See refs. 10-23, cited in Ref. [14]. [18] J.O. Artman, Phys. Rev. 105 (1957) 62, 74. [19] R.I. Joseph, E. Schl6mann, J. Appl. Phys. 38 (1967) 1915.
443
Physica A 241 (1997) 439-443
Origin of linear and nonlinear giant magnetoimpedance A. Yelon*, M. Britel, D. Menard, P. Ciureanu Groupe de Recherche en Physique et Technologie de Couches Minces and Department of Engineering Physics, Ecole Polytechnique, P.O. Box 6079, Station "Centre Ville", Montreal H3C 3A7, Canada
Abstract
It has gradually become clear that giant magnetoimpedance, which initially appeared very mysterious, can be explained classically, as arising from the dependence of skin depth upon the magnetic permeability. As the permeability may vary with dc field or with frequency, the impedance will vary as a function of these parameters. Thus, a variety of circumstances may give rise to GMI. A number of these circumstances are considered. The equivalence of GMI and ferromagnetic resonance calculations is discussed. On the basis of this equivalence, some effects are predicted.
PACS: 72.15.-v; 75.15.Nj; 76.50.+g Keywords: Giant magnetoimpedance; Ferromagnetic resonance; Metal permeability
During its brief history [ 1-6] the phenomenon of giant magnetoimpedance (GMI) has aroused considerable interest, associated with the phenomenon itself, and with its potential applications [6]. The reasons for this interest are easily understood. If we consider a magnetic fiber, wire, or ribbon, to which we apply an alternating current, its impedance is a function of the applied field and of the frequency and amplitude of the current,
Z(H, co,I ) = R + j X .
(1)
The magnetoimpedance may be defined as ~(H,~,I)=
I Z I ( H ) - ]Zl(Hr)
IZl(Hr)
(2) '
is the reference field, usually zero field. In the MHz frequency range, for very soft magnetic materials, positive magnetoimpedance in excess of 100%, [4,7] and w h e r e Hr
* Corresponding author. Fax: 514 340 3218; e-mail: [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 1 2 1-0
A. Yelon et al./Physica A 241 (1997) 439-443
440
negative magnetoimpedance of as much as 70% [2,8] have been observed at low and high values of H, respectively. It has been recognized [2,9,10] that these large changes are due to the change in skin depth arising from the change in magnetic permeability of the material with applied magnetic field. Since the ac current flows in the skin depth, the impedance increases as the permeability increases [11],
~=[
pc2 ] 1/2 L2x~o/~tj
(3)
,
where p is the resistivity, and ~t is the transverse permeability. In order for the effect to be appreciable, the sample must be much thicker than 6 (however, it has been observed in multilayer film structures [6]). In that case [11], Z=(1
- J)2-F~"pL
_
(1 _._~J)L(2~PC°I~t)'/2
(4)
In Eq. (4), L is the length of the sample, and [ is its width, for a ribbon, or a dimension of the order of the circumference for a fiber or wire. We see from Eq. (4) that GMI is favored by large values of p and ~o, and by large changes of #t with H. Thus, the GMI problem is the problem of identifying and quantifying the circumstances in which these conditions, especially the last, are met. In order to investigate these conditions, it is useful to draw a distinction between linear and nonlinear impedance and linear and nonlinear GMI effects. By linear impedance, we mean that Z is independent of I, up to some maximum value, as shown in Fig. 1. Thus, the response to sinusoidal current is a sinusoidal voltage, as shown in Fig. 2(a). This behavior is typical of observations at large values of H, in which the static magnetization is saturated. At fields near, or below those necessary to reverse the magnetization, the typical behavior will be nonlinear, as shown in Fig. 2(b). Typically, linear phenomena have been investigated before nonlinear phenomena, which may be expected to be more complicated. In the development of GMI studies, 10 ........
8-
f
I-+-,0o k.il
........
[
........
A
C~ 6 N
4-
20 0.01
I!
i I
I I
0.1
1
10
I rms (mA)
Fig. 1. Amplitude of the impedance, Z, as a function of the amplitude of the ac current, measured on a 125 lam, as quenched, CoFeSiB wire, at H = 0 , f = 100 kHz.
A. Yelon et al. IPhysica A 241 (1997) 439-443
441
b)
a)
Fig. 2. Current and voltage vs time, measuredon the same sample as for Fig. 1, at lrms= 10mA, f = 1MHz. (a) H = 40 Oe. The current and voltage waveforms are both sinusoidal. (b) H = 0. The current waveform is sinusoidal, but the voltage waveformis not.
it has been the low H, nonlinear effects which have received more attention than the high H, linear effect, despite the fact that they are indeed more complicated. This has been driven by the interest in the extremely high sensitivity obtainable for sensors, or other applications, using low field, positive, GMI. The models which have been used to obtain /tt have frequently been quasistatic in nature. For example, critical curves were used by Beach and colleagues [12] for studying plated wire, and by Humphrey [13] for solid magnetic wire. Machado and Rezende [10] used a domain switching model to treat the behavior of ribbons. These calculations give a good account of experiment up to a few MHz, but it is clear that it is preferable to use a fully dynamic model. Beach and Berkowitz [9] have presented a phenomenological dynamic approach to GMI, and Panina and colleagues [7] have recently developed an expression for #t which takes account of magnetic damping. However, we have recently pointed out [14] that a rigorous theoretical framework which takes account of both magnetic damping and of the exchange conductivity effect was developed over 40 years ago [15,16]. The general theoretical problem of GMI is the determination of the impedance from Maxwell's equations and the Landau-Lifschitz equation, taking account of the conductivity and of the detailed magnetic structure. In the problem of ferromagnetic resonance (FMR) in metals, one needs to find the surface impedance [15] rather than the bulk impedance. However, it is very simple to show [14] that as long as the sample length
442
A. Yelon et al./Physica A 241 (1997) 439-443
is shorter than the electromagnetic wavelength, the two impedances are directly proportional to each other. Since the magnetoimpedance involve a ratio, predictions may be obtained directly from the FMR literature. Since the great majority of FMR experiments are performed at high, saturating, values of H and at low ac fields, there is an abundant literature on high field, linear, calculations and their predictions for metallic samples [17]. In addition, the behavior of metallic samples has been shown not to differ greatly from that of nonmetallic samples (that is, the exchange-conductivity effect modifies the behavior only slightly). Thus for cases for which a rigorous solution is not available, results for insulators yield a good first approximation of what should be expected. There also exist [18] calculations of FMR in multidomain samples which should be useful for those who wish to treat the nonlinear GMI effects observed under these circumstances. Recognition of the simple relationship between GMI and FMR permits us to make predictions for GMI behavior in simple circumstances for which the results are well known. For example, at high fields, the imaginary part of Z should be small, except near FMR. Large values of X at low frequency indicate non-saturation within the skin depth. The dependence of Z on frequency should follow the variation of the real part, as we have observed for permalloy wires [14]. The case of longitudinal GMI (H parallel to 1) in ribbons corresponds exactly to the problem of parallel resonance in plates and films. In fibers and wires, the behavior should be similar. Transverse GMI in a ribbon will be very different, depending upon the orientation of H. H perpendicular to the ribbon corresponds essentially to the problem of perpendicular resonance in plates and films, whereas there should not be a response at large H and small 1, when H is in the plane of the ribbon. However, at large I, there must be a nonlinear response, corresponding to the problem of parallel pumping in FMR [19]. Transverse GMI in fibers and wires must show a relatively small, broad peak, since different sections of the wire have different orientation of H with respect to the radius. What is the future of GMI research, now that the underlying pheonmena are reasonably well understood? Clearly, the range of possible applications is vast [6], and much work needs to be done to optimize materials and geometries for these applications. We also foresee the potential for GMI to become a technique for the study of magnetic materials whose importance should be comparable to, or even exceed, that of FMR. References [1] L.V. Panina, K. Mohri, Appl. Phys. Lett. 65 (1994) 1189. [2] L.V. Panina, K. Mohri, K, Bushida, M. Noda, J. Appl. Phys. 76 (1994) 6198. [3] R.S. Beach, A.E. Berkowitz, Appl. Phys. Lett. 64 (1994) 3652. [41 K.V. Rao, F.B. Humphrey, J.L. Costa-Krfimer, J. Appl. Phys. 76 (1994) 6204. [5] M. Knobel, M.L. Sanchez, J. Velazquez, M. Vasquez, J. Phys.: Condens. Matter 7 (1995) Ll15. [61 L.V. Panina, K. Mohri, T. Uchiyama, Physica A 241 (1997) 429, these Proceedings. [7] L.V. Panina, K. Mohri, T. Uchiyama, M. Noda, K. Bushida, IEEE Trans. Magn. 31 (1995) 1249. [8] P. Ciureanu, P. Rudkowski, G. Rudkowska, D. Mrnard, M. Britel, J.F. Currie, J.O. StriSm-O|sen, A. Yelon, J. Appl. Phys. 79 (1996) 5136.
A. Yelon et al. IPhysica A 241 (1997) 43~443
[9] R.S. Beach, A.E. Berkowitz, J. Appl. Phys. 76 (1994) 6209. [10] F.L.A. Machado, S.M. Rezende, J. Appl. Phys. 79 (1996) 6558. [11] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [12] R.S. Beach, N. Smith, C.L. Platt, F. Jeffers, A.E. Berkowitz, Appl. Phys. Lett. 68 (1996) 2753. [13] F.B. Humphrey, private communication. [14] A. Yelon, D. Menard, M. Britel, P. Ciureanu, Appl. Phys. Lett. 69 (1996) 3084. [15] W.S. Ament, G.T. Rado, Phys. Rev. 97 (1955) 1558. [16] J.R. MacDonald, Phys. Rev. 103 (1956) 280. [17] See refs. 10-23, cited in Ref. [14]. [18] J.O. Artman, Phys. Rev. 105 (1957) 62, 74. [19] R.I. Joseph, E. Schl6mann, J. Appl. Phys. 38 (1967) 1915.
443